By Riaz A Usmani

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Fortunately it can be regularized by a procedure we will refer to as geometric regularization as described in Section 3 and this leads to the numerical invariant R(X, g, ω) from the title, cf. Theorem 1 below. This A Riemannian Invariant, Euler Structures . . 39 invariant for X = − grad f , f a Morse function was considered in [1] in terms of currents. One can also extend the invariant (ii) to vector ﬁelds with isolated zeros, cf. Section 3. A pleasant application of the invariant R and of the extension of (ii) is the extension of the Chern–Simons class from a pair of two Riemannian metrics g1 and g2 to a pair of two smooth triangulations τ1 and τ2 or to a pair of a Riemannian metric g and a smooth triangulation τ , cf.

Thus these functions are multipliers of Cr∗ (Γ) and the corresponding operators Mφr −φr,n are such that Mφr −φr,n ≤ CKn → 0, as n → ∞. 32 J. A. Niblo Since Mφr − Mφr,n = Mφr −φr,n → 0 as n → ∞. This implies that Mφr,n → Mφr = we have Mφr − Mφr,n φr (e) = 1. To get the correct bound on the norm of these operators we introduce scaled functions: 1 φr,n . ρr,n = Mφr,n The algebraic identity satisﬁed by the multipliers, as stated in (2), guarantees that on λ(CΓ) we have the following identity 1 Mφr,n .

A brief survey of approximation properties The study of approximation properties was initiated by Grothendieck in relation to the notion of nuclearity that he introduced in [7]. His fundamental ideas have been applied to the study of groups; in this case one discovers that important properties of groups, like amenability or exactness, can be expressed in terms of approximation properties of the associated operator algebras introduced in the previous section. We give here a brief overview of the main facts.